A farmer with a field along a river wants to form a rectangular enclosure with area 1600 m^2. There is no fence along the side that runs along the river. Determine the minimum length of fencing required by the farmer.

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The farmer wants to create a rectangular enclosure that has an area of 1600 m^2. The enclosure has to be fenced on three sides as the river runs along one of the sides and no fence is required here.

Let the length of the side along which the river runs...

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The farmer wants to create a rectangular enclosure that has an area of 1600 m^2. The enclosure has to be fenced on three sides as the river runs along one of the sides and no fence is required here.

Let the length of the side along which the river runs be L. The length of the other side of the rectangle is 1600/L. The length of the fence required is F = 2*(1600/L) + L

=> F = 3200/L + L

To minimize the length of the fence solve F' = 0

=> `-3200/L^2 + 1 = 0`

=> `L^2 = 3200`

=> `L = sqrt(3200) = 40*sqrt 2`

The length of the fence is `40*sqrt 2 + 40*sqrt 2 = 80*sqrt 2`

The minimum length of the fence is `80*sqrt 2` m.

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